Question Paper: Applied Mathematics 2 : Question Paper May 2016 - First Year Engineering (Semester 2) | Mumbai University (MU)
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## Applied Mathematics 2 - May 2016

### First Year Engineering (Semester 2)

TOTAL MARKS: 80
TOTAL TIME: 3 HOURS
(1) Question 1 is compulsory.
(2) Attempt any three from the remaining questions.
(3) Assume data if required.
(4) Figures to the right indicate full marks.
1(a) Prove that $$\int ^1_0\frac{dx}{\sqrt{-\log x}}=\sqrt{\pi}$$(3 marks) 1(b) Solve $$\dfrac{d^3y}{dx^3}-5\dfrac{d^2y}{dx^2}+8\dfrac{dy}{dx}-4y=0$$(3 marks) 1(c) Prove that Δ∇=Δ&nabla(3 marks) 1(d) Solve [xy sin(xy) + cos (xy)] y dx+[x ysin(xy) - cos (xy)]x dy=0(3 marks) 1(e) Change to polar coordinates and evaluate $$\displaystyle \int ^1_0\int ^{\sqrt{2x-x^2}}_x \left ( x^2+y^2 \right )dx\ dy$$(4 marks) 1(f) Evaluate $$\int ^1_0\int ^x_0\left ( x^2+y^2 \right )xdy\ dx$$(4 marks) 2(a) Solve $$\left ( 1+y^2 \right )dx=\left ( e^{\tan^{-1}y}-x \right )dy$$(6 marks) 2(b) Change the order of integration and evaluate $$\int ^1_0\int ^{\sqrt{1-x^2}}_0\frac{e^y}{\left ( e^y+1 \right )\sqrt{1-x^2-y^2}}dy\ dx$$(6 marks) 2(c) Prove that $$\int ^{\infty}_0\dfrac{e^{-x}-e^{-ax}}{xsecx}dx=\dfrac{1}{2}\log\left ( \dfrac{a^2+1}{2} \right )$$(8 marks) 3(a) Evaluate $$\int ^e_1\int ^{\log y}_1\int ^{e^x}_1\log z\ dz\ dy\ dx$$(6 marks) 3(b) Find the total area od the curve r = a sin2?(6 marks) 3(c) Solve $$x^2\dfrac{d^3y}{dx^3}+3x\dfrac{d^2y}{dx^2}+\dfrac{dy}{dx}=x^2\log x$$(8 marks) 4(a) Show that thw length of the arc of the curve ay2 = x3 from the origin to the point whose abscissa is b is $$\dfrac{8a}{27}\left [ \left ( 1+\dfrac{9b}{4a} \right )^{3/2} -1\right ]$$(6 marks) 4(b) Solve $\left ( D^2-D-2 \right )y=2 \log x+\dfrac{1}{x}+\dfrac{1}{x^2}$(6 marks) 4(c) Apply Runge-Kutta Method of fourth order to find an approximate value of y for $\dfrac{dy}{dx}=x\ y$ with x0=1, y0 =1 at x=1.2 taking h=0.1(8 marks) 5(a) Solve $\left ( x^2y-2xy^2 \right )dx-(x^3-3x^2y)dy=0$(6 marks) 5(b) Using Taylor series Method obtain the solution of following differential equation $\dfrac{dy}{dx}=2y+3e^x$ with y0 = 0 when x0 = 0 for x=0.1, 0.2(6 marks) 5(c) Find the approximate value of $\displaystyle \int ^4_0 e^x dx$
by i) Trapezoidal Rule, ii) Simpson's 1/3rd Rule
(8 marks)
6(a) In a circuit containing inductance L, resistance R, and voltage E, the current I is given by $L\dfrac{di}{dt}+Ri=E.$ Find the current i at time t if at t=0, i=0 and L, R, E are constants.(6 marks) 6(b) Evaluate $\iint _R\dfrac{dxdy}{\left ( 1+x^2+y^2 \right )^2}$ over one loop of the lemniscate $\left ( x^2+y^2 \right )^2=x^2-y^2$(6 marks) 6(c) Find the volume bounded by the cylinder x2 + y2 = 4 and the planes z=0 and y+z=4(8 marks)